A267816 - OEIS

%I #20 Apr 16 2019 15:25:15

%S 1,3,23,111,479,1983,8063,32511,130559,523263,2095103,8384511,

%T 33546239,134201343,536838143,2147418111,8589803519,34359476223,

%U 137438429183,549754765311,2199021158399,8796088827903,35184363700223,140737471578111,562949919866879

%N Decimal representation of the n-th iteration of the "Rule 221" elementary cellular automaton starting with a single ON (black) cell.

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A267816/b267816.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjectures from _Colin Barker_, Jan 22 2016 and Apr 16 2019: (Start)

%F a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.

%F G.f.: (1-4*x+16*x^2-16*x^3) / ((1-x)*(1-2*x)*(1-4*x)).

%F (End)

%F a(n) = 2^(n + 1) * (2^n - 1) - 1, for n > 0. - _Jaroslav Krizek_, Jan 22 2017

%t rule=221; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

%Y Cf. A267814.

%Y Similar entries: A085601 (2^(n + 1) * (2^n + 1) + 1), A092431 (2^(n - 1) * (2^n + 1) - 1), A092440 (2^(n + 1) * (2^n - 1) + 1), A129868 (2^(n - 1) * (2^n - 1) - 1), A134169 (2^(n - 1) * (2^n - 1) + 1), A281481 (2^(n - 1) * (2^n + 1) + 1), A281482 (2^(n + 1) * (2^n + 1) - 1). - _Jaroslav Krizek_, Jan 22 2017

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 20 2016